cp's OEIS Frontend

A106515 A Fibonacci-Pell convolution.

%I A106515 #26 Feb 08 2025 14:55:15
%S A106515 1,2,6,15,38,94,231,564,1372,3329,8064,19512,47177,114010,275430,
%T A106515 665247,1606534,3879302,9366735,22615356,54601628,131825377,318263328,
%U A106515 768369744,1855031473,4478479058,10812064614,26102729679,63017720390
%N A106515 A Fibonacci-Pell convolution.
%C A106515 Diagonal sums of A106513.
%H A106515 G. C. Greubel, <a href="/A106515/b106515.txt">Table of n, a(n) for n = 0..1000</a>
%H A106515 Tamás Szakács, <a href="https://hdl.handle.net/2437/381856">Linear recursive sequences and factorials</a>, Ph. D. Thesis, Univ. Debrecen (Hungary, 2024). See pp. 28, 51, 58.
%H A106515 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-3,-1).
%F A106515 G.f.: (1-x)/((1-x-x^2)*(1-2*x-x^2)).
%F A106515 a(n) = Sum_{k=0..n} Fibonacci(n-k-1)*Pell(k+1).
%F A106515 a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..floor((n-k+1)/2)} binomial(n-k+1, 2*j+k+1)*2^j.
%F A106515 a(n) = Pell(n) + Pell(n+1) - Fibonacci(n). - _Ralf Stephan_, Jun 02 2007
%F A106515 a(n) = 3*a(n-1) - 3*a(n-3) - a(n-4). - _Wesley Ivan Hurt_, May 27 2021
%t A106515 Table[Fibonacci[n, 2] + Fibonacci[n+1, 2] - Fibonacci[n], {n, 0, 30}] (* _Vladimir Reshetnikov_, Sep 27 2016 *)
%t A106515 LinearRecurrence[{3,0,-3,-1},{1,2,6,15},30] (* _Harvey P. Dale_, Feb 08 2025 *)
%o A106515 (Magma)
%o A106515 Pell:= func< n | Round(((1+Sqrt(2))^n - (1-Sqrt(2))^n)/(2*Sqrt(2))) >;
%o A106515 [Pell(n) + Pell(n+1) - Fibonacci(n): n in [0..30]]; // _G. C. Greubel_, Aug 05 2021
%o A106515 (Sage) [lucas_number1(n+1, 2, -1) + lucas_number1(n, 2, -1) - lucas_number1(n, 1, -1) for n in (0..30)] # _G. C. Greubel_, Aug 05 2021
%Y A106515 Cf. A000045, A000129, A106513.
%K A106515 easy,nonn
%O A106515 0,2
%A A106515 _Paul Barry_, May 05 2005